oh5 - LECTURE 5: The Classical Linear Regression Model...

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LECTURE 5: The Classical Linear Regression Model Suppose that we are interested in estimating 1 and 2 in the following model: Y i = 1 + 2 X i + " i We may estimate the unknown 1 and 2 by OLS, that is, by forming ^ 1 = Y ^ 2 X ^ 2 = P i X i X Y i Y ± P i X i X ± 2 If we want to say something about the stochastic properties of the OLS estimators of 1 and 2 (for example, to calculate their bias, about how the error terms are generated in the PRF. The following assumptions constitute the Classical Linear Regression Model (CLRM) which is also known as the Simple Linear Regression Model (SLRM). : 1
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ASSUMPTIONS OF THE CLRM (aka SLRM): 1. Linearity in the parameters. Y is generated as: Y i = 1 + 2 X i + " i (PRF) 2. X i repeated samples): OR If X is stochastic, then, in the sample not all X i ±s are equal to some constant 3. The expected, or mean, value of the disturbance term " i is zero (If X is nonstochastic): E ( " i ) = 0 OR The error term " has zero conditional mean given X : E ( " i j X i ) = 0 4. The variance of each " i is constant for all i , that is, " i is homoskedas- tic : If X is nonstochastic: var ( " i ) = ± 2 OR If X is stochastic: 2
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( " i j X i ) = 2 . 5. There is no correlation between two error terms. This is the as- sumption of no autocorrelation or no serial correlation : cov ( " i ; " j ) = 0 8 i 6 = j This will also be true if we assume:that the pairs ( X i ; Y i ) are indepen- dent and identically distributed across i = 1 ; :::; n: This assumption is equivalent to assuming that ( X i ; " i ) are i.i.d. FROM NOW ON, WE WILL ASSUME THE Xs ARE NON-
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oh5 - LECTURE 5: The Classical Linear Regression Model...

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